Monday, June 7, 2010

On Undecidable Propositions

So... maybe I'm not as done talking about Professor Kuspit as, at first, I thought. I'm reading his book, The End of Art, which is very interesting. I'd like to discuss a point he raises, but given my absolutely crap track record for interpreting him correctly, let me say that any text not actually in quotation marks does not necessarily represent Kuspit's thinking. Even if I say it does. Just take it with a grain of salt, is all I'm saying.

Kuspit devotes a lot of time and energy to having it out with Marcel Duchamp. Duchamp, you may remember, is the fellow who gave us this:

That's the "readymade," the famous urinal converted - presto - into art on Duchamp's say-so. It's been all downhill for art since then, basically. Here's what caught my eye in Kuspit's discussion:

"Clearly the readymade has a double meaning. It is a conundrum, a Gordian knot that no intellectual sword can cut. Simultaneously an art and non-art object, the readymade has no fixed identity. Regarded as art, it spontaneously reverts to non-art. It collapses into banality the moment the spectator takes it seriously as art, and becomes serious art the moment the spectator dismisses it as a banal object. Just as the spectator critically reacts to it, thinking about and looking at it in a more creative way than he thinks about and looks at non-art objects, it becomes one of those non-art objects. The readymade always outsmarts the spectator, outwitting his interpretation of it ... it...remains indecipherable. It is absurd..."

(p. 23)

Now, there's nothing I like more than an ontologically ambiguous phenomenon! And who doesn't, really? Sometime, we'll talk about De Chirico. But not today.

Frankly, I had never given much thought to Duchamp's readymades before, because basically, it's a goddamn urinal, so who cares? I had never experienced this frisson of the irresolvable that Kuspit describes. But his description of this particular frisson was immediately familiar. He describes an object which, when contemplated, triggers an endless oscillation between two unstable states, the acceptance of either of which immediately causes a collapse of the state and a recoil to the opposed state. Why is that familiar? Because it is exactly the same frisson triggered by the following sentence:

This statement is a lie.

Think about it for a second, you'll see what I mean.

The urinal dates to 1917. The sentence comes to our attention in 1931. It is the plain-language example of Kurt Gödel's first incompleteness theorem. Imagine a theory, called T. A slightly less plain-language version of the statement is called "sentence G for the theory T," or the Gödel sentence:

G cannot be proved to be true within the theory T.

The incompleteness theorem is a devastating discovery which effectively ended a program, championed by Bertrand Russell, David Hilbert, and the adorable Gottlob Frege, to put mathematics on a really solid footing by finally demonstrating the completeness and consistency of certain absolutely necessary and fundamental mathematical propositions.

"Completeness" and "consistency" have special meanings here. A "complete" system is capable only of generating statements that can be proved or disproved inside the system. A "consistent" system is incapable of generating statements that can be both proved and disproved in the system. Got that?

1. Completeness: capable of generating only statements that can be either proved or disproved within the system

2. Consistent: incapable of generating statements that can be both proved and disproved within the system

For the system of fundamental mathematics to live up to the hopes of Russell, Hilbert, and Frege, it must be both complete and consistent. What Gödel does is to prove that *no* system capable of generating a theory of fundamental mathematics can be both complete and consistent. If it is complete, it is inconsistent. If it is consistent, it is incomplete.

Incidentally, you haven't missed how Gödel proves this, because I haven't described it here. It's fiendishly complicated and involves a lot of notation that you'd probably be happier not knowing. It's a short book, if you're really interested. A short book, but if you're anything like me, a long fricking read.

What's interesting for our purposes here is that Kuspit's description of his experience of looking at Duchamp's urinal is categorically identical with the experience of looking at the deadly Gödel sentence. Given that, let us entertain for a moment the possibility that the Duchamp urinal is a Gödel sentence. Wow! What would that mean?

It would mean that art, when looked at from a certain perspective, can be functionally considered to be a formal system of logic. It would mean that the urinal is a statement, consistent with the language of this system, which makes an assertion that is undecidable within the system.

Well, what is the character of this system? And what assertion is the urinal making?

A system is a set: it is a set of axioms and theorems. One really important property of sets is that they should define what they include and don't include. How is the set defined? This property, the formation of the boundary of the set, allows the analyst to distinguish between relevant and irrelevant phenomena when conducting an analysis of the set. For instance, "1 + 1 = 2" is an important member of the set of fundamental mathematics. "Meow" is not.

In the art context, defining the boundaries of the set amounts to answering the question: What is art?

Whoa! Holy shit! The big question!

What Duchamp has done, in proposing the urinal as a work of art by signing it, is to point out a contradiction implicit in the set of axioms which underlie the definition of art. Axioms are statements taken to be true (though not proven) at the base of a logical system. Good axiom selection involves picking axioms that generate a minimum number of inconsistent results (theorems).

Well, art has a little axiom problem with its boundary definition. Duchamp has figured out that we have at least two axioms determining the boundary of the set we call art:

Axiom 1: "Art" consists of that set of human-produced objects which use aesthetic means to produce some kind of enlightening transformation in the viewer.

Axiom 2: "Art" consists of that set of objects produced by people called "artists" who come up to you and tell you that the thing they just made is "art."

Feel free to re-interpret axiom 1 as you like, depending on your personal tastes. Axiom 2 you'd have a hard time arguing with in any consistent way.

Duchamp, noticing the not-necessarily-identical outputs of axiom 1 and axiom 2, has gone ahead and generated an object which, under axiom 1, is "not-art," and under axiom 2, is "art." Because both axioms are deeply and viscerally active to the modern art viewer, Duchamp's object produces a sense of rapid and unstable oscillation between art and non-art states, as described by Kuspit: it is a contradictory object, a basilisk, a Gödel sentence.

Any artist could have done this at any time, because this problem has always existed in art and is intuitively obvious to any working artist. But before Duchamp, there had kind of been a tacit agreement among artists not to be a total dick about it. Functionally speaking, the artists would exploit axiom 2, while ensuring that they did their best not to violate axiom 1.

Duchamp's innovation lies not in his insight, but in the audacity of his barbarity. Just because you're the first rapist doesn't mean nobody thought of committing rape before. They just had the sense not to do it. Duchamp's program is not constructive, in the sense of seeking to highlight logical problems in order to point a way toward resolving them, as best as possible. His program is destructive; he highlights problems in order to wreck trust in the relevant system. In so doing, he discredits everything previously generated by the system, and shuts down subsequent use of the system.

In a sense, it is a salutary effect, because it forces us from the age of naive art-making to the age of willful art-making, with regard to that aspect of art which corresponds with a logical system. You might call the urinal one of my prime numbers, even though it is not, in itself, productive. And you would do well not to forget this: art is not a logical system.

However, the logical system aspect of art is obviously an important part of it, and since the urinal, a good deal of speculation on the axiomatic problem - What is art? - has gone on. I myself am not particularly a philosophical purist, so I like Roger Kimball's rough-and-ready solution to the problem:

"...the real issue is not whether a given object or behavior qualifies as art but rather whether it should be regarded as good art. In other words, what we need is not definitional ostracism but informed and robust criticism."

(p. 53, Art's Prospect)

You see what he does here? He demotes Axiom 1 - the "aesthetics and transformation" axiom - from its status as a definer of boundaries. He leaves only axiom 2 to define the set of objects called "art." But having done so, he immediately re-applies axiom 1 as an axiom, not of definition, but of criticism.

That works for me, but I tell you what. I described this idea to my wife, Charlotte, who is nothing if not skeptical, and she said, "That's really begging the question, isn't it? I mean, if you want to focus on what good art is, you're presupposing an answer to the question of what art is, aren't you?"

More on Charlotte's opinions soon. In the meantime, here's a huge-ass painting I just finished that I am very happy with, because any time you get to use a lot of cobalt blue, you can't help walking away happy:

Rachel's Room
2010, oil on canvas, 72"x48"


  1. Interesting article, Doktor. This Kuspit frisson reminds me somewhat of the vertigo that I get when I walk down an escalator that is stationary. My mind has difficulty resolving that the moving steps aren't moving and my equilibrium is affected.

    "But before Duchamp, there had kind of been a tacit agreement among artists not to be a total dick about it."

    This is founded on the fact that no one before Duchamp signed a pisser? I suspect that we have far more biographical evidence – pre-Duchamp – of artists being total dicks than we do of the existence of a tacit agreement of this sort. Just because you silently signed this pact certainly doesn't mean most everyone else did. Most people are too simple.

  2. Kelly - I hope you will not go falling down any escalators! It would be comical, but also, it would end in tears.

    I think we can agree that artists have a long and storied history of being total dicks. Just not so much of a history of them being dicks about the inconsistencies resulting from conflicts between axioms 1 and 2. However, I do have to concede your point - it is possible that some of them *were* trying to be dicks about that too. They just didn't get as much press as Duchamp.

    Kuspit suggests in the book that it really chapped Duchamp's ass that Matisse and Picasso were much more successful than he was. So, chess player that he was, he deployed the Godel gambit on them and their fancy pictures. I recommend Kuspit's book - it's really interesting.

    I did silently sign the pact, as you say. I'm not sure anyone has phrased the pact in this particular way before, and I suspect that most people either sign it, or don't sign it, in a kind of intuitive way that doesn't involve so much citing of mathematicians. I do imagine Duchamp had a fairly clear idea what he was doing, although of course, in 1917 he couldn't possibly have phrased it in terms of Godel's 1931 work.

  3. That's some clever bit of reasoning there! It kind of sends you into an infinite loop between axioms. The Urinal says, "I am not a piece of art." And you say, "But I am viewing you as one, so you must be." And the Urinal says, "But I'm not." And on and on.

    Didn't Kirk and Spock once use Godel's statement to defeat an evil robot?

    You're right about Duchamp and chess. He played some world class people in the 20s and 30s and even beat some of them. Maybe it helped him to accept his lack of success compared to Picasso and Matisse knowing that he could crush them on the board even after giving them piece odds and the move.

  4. Chris - I think the problem is not so much that, as that the urinal is saying to you, "I am art," at the same time that it is saying, "I am not art," and you are saying the same two things to it.

    I don't know about Kirk and Spock, but I do know that the ghost of Dave Bowman, psychically fused with the ghost of Hal, uploads a virus based on Godel's work to the malfunctioning Monolith in "3001: The Final Odyssey." Arthur C. Clarke was horribly embarrassed when "Independence Day" came out around the same time with approximately the same ridiculous plot point.

    Duchamp's chess playing implies, to me, an analytic outlook which would have satisfied him not only on the egotistical level you describe, but also would have provided him with a) a strategic approach to his war with Picasso and Matisse and b) a rationalistic outlook on art which is not necessarily congruent with making good art. Of course, he might have made art we object to for reasons unrelated to that; he might have pursued the discipline of strongly segregating "art" and "reasoning" brain functions, a discipline I myself pursue.

    That said, I would like to think that, mano-a-mano, Kubrick would have made him look like a woodpusher.

  5. You don't remember the scene on that planet of androids where Spock causes a computer to self-destruct by giving it a contradictory statement? I think the statement was, "I am telling a lie."

    Alas, I believe Duchamp would have crushed Kubrick. Duchamp has wins and draws over top international competition including women's world champ Vera Menchik. He also came within a whisker of becoming the French champ. According to, Kubrick never made it to that level.

  6. I am ashamed to admit it, but I do not know that episode. Seems like some sloppy programming if they left the computer vulnerable like that. Let me guess, though: Kirk takes a breather from the main plot to teach a hot female android about the unquantifiable nature of love.

    Yeah, OK, maybe Duchamp would have crushed Kubrick. But man, I would have been rooting for the wry humanist over the sarcastic anti-humanist.

  7. Kubrick? A humanist? This is the guy who directed Lolita and the Shining and Eyes Wide Shut and A Clockwork Orange. Are you sure you would call him a humanist? Did you not see the ending of Full Metal Jacket? Are you not aware that he destroyed the world at the end of Dr. Strangelove? I think Duchamp and Kubrick share much by way of subversiveness and irony, but Kubrick focused on subverting or satirizing humanity whereas Duchamp focused on subverting and satirizing art. One big difference is that Duchamp's work is bolstered by a clever but relatively cheap intellectualism whereas Kubrick's is bolstered by genius.

    When you called Kubrick a wry humanist perhaps you were getting him mixed up with Vittorio De Sica? Or maybe Ron Howard. That's it. You must have been thinking about Ron Howard. I didn't know Ron Howard played chess.

  8. Look, there's a different between an anti-humanist and an embittered humanist. I've always taken Kubrick for the latter. He seems to me deeply sensitive to cruelty, even fascinated by cruelty, but always in the context of a shattered disappointment in what men make of their lot. Just because he can laugh at suffering, and accept the inevitability of suffering, doesn't make him less outraged at suffering, and thus, outraged at the indifferent tragedy of the universe.

    But look, this is an *art* blog. Maybe you want to start a *film* blog, and I can come over there and write a bunch of comments. Whaddaya think of that? How does that strike ya?

  9. You will notice I wrote "different" instead of "difference" there. This is because I am an illiterate cretin.

  10. For me there is another aspect to Duchamp's readymades, the urinal and the bicycle wheel and the bottle rack. They are wonderful, aesthetically beautiful things that we just wouldn't even notice if Duchamp had not isolated them and put them on a pedestal. It is art not because Duchamp says so, but because he saw it with the artist's eye. Communicating the experience of the artist's way of seeing something is a huge part of what art is about, and when you see it this way it doesn't violate axiom 1 at all; in fact, it fulfills it perfectly. He selects something and arranges it and makes it sculpture. Is this so different from a fine art photographer who sees something wondrous and uses her craft to make an image that conveys the experience of wonder in the found object?

  11. Fred - this is probably the best defence I've read so far of the readymades. Your juxtaposition of the possible interpretation you raise with the action of the photographer raises a question (and if I've treated this already, please forgive me - I haven't read the post in a while...): To what extent do we demand that an artist somehow remake nature before the artist's work is considered to be an aesthetic expression? For instance, figurative painting is a radical remaking, in the sense that the product (an image of a person) does not particularly resemble the materials (pigment particles mixed with oil, say). Photography has always been on the fine line of this problem, as you know, with some arguing that the transformation produced by picture-taking is sufficient to qualify the picture as art, and others taking the reverse view. Duchamp's action is even more minimal; and the enlightenment you describe may be categorically different from, say, the heightened awareness of moonrise we might derive from a well-written haiku on the subject. But I cannot deny that you make a valid point.

    There is a second question that is worth considering, a sort of cost-benefit question. Does Duchamp cost us, in terms of his destruction of the boundaries and possibilities of art, more than he benefits us, in terms of the awareness of the beauty of the everyday that you describe? Reasoning backward from that, we have another question we must answer first: is it legitimate to ask a cost-benefit question of this type about artwork?

    I don't have any answers, and obviously I come down on the other side of the fence from you on the question of the readymades, but I think your position is the most defensible pro-Duchamp position I have encountered. Thanks for taking the trouble to write it up.

  12. I think any question about what qualifies as art cannot rest on what the medium is. Obviously there are millions of photographers in the world producing gazillions of images. Considering all the photographers to be artists and all the images art would dilute the concept to meaninglessness. At the same time it would be hard to deny that certain photographers, even if they are essentially using technological means to capture images of found reality, are in fact artists, conveying a transformative vision through highly developed craft. Henri Cartier-Bresson, for instance, may have been a snapshooter, but he created a powerful body of work that I expect will be treasured and appreciated for centuries to come.

    By the same token, there are a lot of figurative painters and drawers out there, and most of them, even if they have highly developed technique, aren't really doing anything very interesting. We can call these people artists as a generic term, but what's worthwhile as art surely has to be based on something other than the medium or the subject matter or even the craftsmanship.

    Duchamp of course did many things besides the readymades, which makes it possible to see them in the context of a whole body of objects and ideas. To me it is clear that he was a joker - a brilliant wit and a trickster. Unfortunately many artists and art critics have no sense of humor. Both Duchamp's detractors and his admirers often seem to mistake the joker for a bomb-throwing anarchist.

  13. Fred -

    I am ambivalent about photography myself, but I can't deny that I consider some photographs to be art. The restriction of what is called art by medium is an interesting problem, and while I think we would all have been better off agreeing on a classical menu + film and leaving it at that, obviously that is neither philosophically defensible nor practical. So we confront a proliferating series of problems. The problem of which photographs are and are not art, and which paintings are and are not art, is a very good problem. My wife Charlotte contends that there is no such thing as bad art; that any art that does not fulfill the more abstract criteria of art is simply not art at all. But this locates the problem in the realm of criticism, about which we are all even less likely to agree than we are about definitional formulations. So it's a big can of worms, as far as I can tell. If you want to elaborate some more on the non-medium/subject/craft criterion, I am very interested in your thoughts, even if I don't have anything particularly to contribute at this juncture (it's not a problem that's been absorbing me lately, so I'm just kind of replying to you here by saying, "Yeah, good point.")

    As for Duchamp, you present a different and, once again, impressive defense of his position. I agree with you that it's worthwhile to consider his readymades in the context of his entire output, but I disagree with your final thought in the following sense: It is entirely possible for a joker to throw a bomb.

    1. Maybe we should take Duchamp at his word: It is not art. Someone has called work of this kind "concrete philosophy," and I think your analysis supports this rubric.

  14. Too late! Duchamp's little joke has gotten out of control, it seems, and, at least insomuch as we want to be philosophically responsible so we can sleep soundly at night, I think we're obliged to sort through the century of wreckage resulting from Duchamp's bomb. Although if you can place the work as you describe it, and be content - I don't think there's anything wrong with that, and admire you for it.